The surface area of a rectangular pyramid is calculated by adding the areas of all the faces of the pyramid. In this type of pyramids, we have one rectangular face and four triangular faces. The opposite triangular faces have the same area. In total, we need three lengths to calculate the surface area of these pyramids: the width and base of the rectangle and the slanted height of one of the triangular faces.

Here, we will learn about the formula for the surface area of rectangular pyramids. Also, we will use this formula to solve some practice problems.

## Formula to find the surface area of a rectangular pyramid

The surface area of rectangular pyramids is equal to the sum of the areas of all the faces of the pyramid. These pyramids are composed of a rectangular face and four triangular faces. The opposite triangular faces are the same. Therefore, we have to find expressions for the areas of triangles and rectangles.

Remember that the area of a rectangle is equal to the length of its base multiplied by its width. Therefore, for the base, we have the area $latex A=ba$, where *b* is the length of the base and *a* is the length of the width.

On the other hand, we know that the area of a triangle is equal to one-half of the triangle’s base multiplied by the height. We have two different bases for the triangles, *b* and *a*, which are the lengths of the rectangular base. Also, the slant height of the triangles is the same.

Therefore, we have two different areas for the triangles, $latex \frac{1}{2} bh$ and $latex \frac{1}{2} ah$, where *h* represents the length of the slant height of the triangular faces. We have two triangles with each of these areas, so the total surface area of the pyramid is:

$latex A_{s}=ba+bh+ah$

## Surface area of a rectangular pyramid – Examples with answers

The following examples are solved using the formula for the surface area of rectangular pyramids. Each example has its respective solution, where the reasoning and process used are detailed.

**EXAMPLE 1**

What is the surface area of a pyramid that has a rectangular base with a width of 5 m and a base of 4 m and where the slant height of the faces is 5 m?

##### Solution

We have the following values:

- Rectangle width, $latex a=5$
- Rectangle base, $latex b=4$
- Triangles slant height, $latex h=5$

Using these values in the formula for surface area, we have:

$latex A_{s}=ba+bh+ah$

$latex A_{s}=(4)(5)+(4)(5)+(5)(5)$

$latex A_{s}=20+20+25$

$latex A_{s}=65$

The surface area is equal to 65 m².

**EXAMPLE 2**

A pyramid has a rectangular base with a width of 6 m and a base of 7 m. If the slant height of the triangular faces is 8 m, what is the surface area?

##### Solution

We recognize the following information:

- Rectangle width, $latex a=6$
- Rectangle base, $latex b=7$
- Triangles slant height, $latex h=8$

We substitute these values in the formula for surface area:

$latex A_{s}=ba+bh+ah$

$latex A_{s}=(7)(6)+(7)(8)+(6)(8)$

$latex A_{s}=42+56+48$

$latex A_{s}=146$

The surface area is equal to 146 m².

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**EXAMPLE 3**

What is the surface area of a rectangular pyramid that has triangular faces with a height of 9 m and a rectangular base with a width of 7 m and a base of 9 m?

##### Solution

We have the following information:

- Rectangle width, $latex a=7$
- Rectangle base, $latex b=9$
- Triangles slant height, $latex h=9$

By substituting these values in the formula for surface area, we have:

$latex A_{s}=ba+bh+ah$

$latex A_{s}=(9)(7)+(9)(9)+(7)(9)$

$latex A_{s}=63+81+63$

$latex A_{s}=207$

The surface area is equal to 207 m².

**EXAMPLE 4**

If a rectangular pyramid has a rectangular base with a width of 8 m and a base of 10 m and the height of its triangular faces is 11 m, what is its surface area?

##### Solution

We have the following information:

- Rectangle width, $latex a=8$
- Rectangle base, $latex b=10$
- Triangles slant height, $latex h=11$

We find the surface area using these values in the formula:

$latex A_{s}=ba+bh+ah$

$latex A_{s}=(10)(8)+(10)(11)+(8)(11)$

$latex A_{s}=80+110+88$

$latex A_{s}=278$

The surface area is equal to 278 m².

## Surface area of a rectangular pyramid – Practice problems

Test what you have learned about the surface area of rectangular pyramids and solve the following problems. Check out the solved examples above in case you need help.

## See also

Interested in learning more about triangular pyramids? Take a look at these pages:

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